practical application of geotechnical limit analysis in limit state design

Jan/Feb 2009 Seminar: Geotechnical Stability Analysis to Eurocode 720/05/2008 LimitState:GEO launch & technology briefing - ICE London geo1.0

Practical Application of

Geotechnical Limit Analysis in

Limit State Design

Dr Colin Smith

University of Sheffield, UK

Director, LimitState Ltd UK

Contents

• Ultimate Limit State Design to Eurocode 7

• Factors of safety

• Use of Partial Factors

• Eurocode 7 : advantages and disadvantages

• Practical examples (II)

Limit State Design

Specific Objectives

• At the end of this lecture , delegates should be

aware of:

– the core principles underlying Limit State Design

– the implementation of Limit State Design as

exemplified by Eurocode 7

– the use of Limit State Design with computational limit

analysis

– challenges and advantages of Limit State Design

Jan/Feb 2009 Seminar: Geotechnical Stability Analysis to Eurocode 720/05/2008 LimitState:GEO launch & technology briefing - ICE London 31/07/2009 geo1.0

Limit state

analysis / design

What is ‘limit state’ analysis / design?

• Limit states are states beyond which the

structure no longer satisfies the relevant design

criteria

– focus on what might go wrong

– e.g. ultimate limit state (ULS) and serviceability limit

state (SLS)

• Not directly related to specific analysis methods

What is ‘limit state’ analysis / design? [2]*

• „An understanding of limit state design can be

obtained by contrasting it with working state

design:

– Working state design: Analyse the expected, working

state, then apply margins of safety

– Limit state design: Analyse the unexpected states at

which the structure has reached an unacceptable limit

– Make sure the limit states are unrealistic (or at least

unlikely)‟

*Brian Simpson, Arup

The ultimate ‘limit state’

• According to EC 0 („Basis of structural design‟):

[Serious consequences of failure mean this must

be rendered a remote possibility]

Implementation in practice

Consider actions (loads) and resistances (i.e. resistance available at collapse)

Implementation in practice [2]

Partial factors are applied to Actions and Resistances

Probability

of

occurrence

Parameter

valueAction Resistance

But…

Should we take the resistance at the footing/soil interface?

Or in the soil?

But… [2]

Jan/Feb 2009 Seminar: Geotechnical Stability Analysis to Eurocode 731/07/2009 geo1.0

Implementation of

Limit State Design

in Eurocode 7

• Eurocode 7 has been under development since 1980.

• Use of Eurocode 7 will be mandatory in the UK in 2010.

– Original draft had 1 Design Approach (DA)

– Now has 3 different Design Approaches. Individual countries are free to adopt one or more approaches for national use

Eurocode 7

Eurocode 7 moves away from:

• working stresses

• overall factors of safety

to:

• limit state design approach

• partial factors

• unified approach

Eurocode 7

It introduces a range of separate checks:

• Ultimate Limit States (ULS)

• Serviceability Limit States (SLS)

Eurocode 7

EQU : loss of equilibrium

STR : failure of the structure

GEO : failure of the ground

UPL : failure by uplift

HYD : hydraulic heave

Existing codes

Existing codes often combine SLS and ULS:

Advantages:

• Simple

• Only one calculation required

Disadvantages:

• Not necessarily transparent,

• „Safety‟ factors typically only applicable for a specific subset of parameters

In the UK the checks against failure in the ground

(GEO) and in the structure (STR) must be checked

using „Design Approach 1 (DA1)‟ which requires that

two checks are performed:

Design combination 1 (DA1/1)

Design combination 2 (DA1/2)

Eurocode 7

Different factors are applied if loads (Actions) are:

• Permanent• Variable• Accidental

Different factors are applied if loads (Actions) are:

• Favourable• Unfavourable

i.e assist or resist collapse (this may not be clear at outset)

Loads (Actions)

EC7 DA1/1 EC7 DA1/2 Conventional bearing

capacity

BS8002 (Retaining

walls)

Permanent

unfavourable load 1.35 1.0 1.01.0

Variable

unfavourable load 1.5 1.3 1.0 1.0

Permanent

favourable load 1.0 1.0 1.0 1.0

c' 1.0 1.25 1.0 1.2

tan' 1.0 1.25 1.0 1.2

cu 1.0 1.4 1.0 1.5

Resistance 1.0 1.0 2.5 – 3.5 N/A

Partial Factors

Quick Guide


Eurocode 7 Example:

Foundation Design

Example – foundation design

Simple design case:

– Footing on undrained (clay soil)

Examine:

– Conventional design approach

– Design to Eurocode 7

Example – foundation design [2]

Terzaghi‟s bearing capacity equation:

cu = 70 kN/m2

= 20 kN/m3

q = 10 kN/m2 *

W = 40 kN/m

B = 2m

BNqNcNBR

qc 21/

V

Wqq

R

B

*Assume q is a permanent load

For the undrained case:

qcBRu )2(/


Design to Eurocode 7Eurocode 7 requires a change in conceptual approach. Instead of finding a collapse load and factoring it, it is necessary to define Actions andResistances in the problem and ensure that

Actions ≤ Resistances

To ensure reliability of the design, Partial factors are applied to the actions and resistances/material properties.


The actions (E) driving the footing to collapse are:

cu = 70 kN/m2

= 20 kN/m3

q = 10 kN/m2

W = 40 kN/m

B = 2m

V

Wqq

R

•Foundation load V

•Foundation self weight W


The Resistance R opposing collapse of the footing is provided by:

cu = 70 kN/m2

= 20 kN/m3

q = 10 kN/m2

W = 40 kN/m

B = 2m

V

Wqq

R

•Soil resistance R (applied at base of footing)


cu = 70 kN/m2

= 20 kN/m3

q = 10 kN/m2

W = 40 kN/m

B = 2m

V

Wqq

Conventional DA1/1 DA1/2

Design Actions (Ed)

E=V+WV+40 1.35(V+40) 1.0(V+40)

Design Resistance (Rd)

R=B(5.14cu+q)

Rd (kN/m) 247 740 534

Ed<Rd V≤207 kN/m V≤508 kN/m V≤494 kN/m

0.3

)107014.5(2

0.1

)104.1

7014.5(2

0.1

)100.1

7014.5(2

Eurocode 7

Comments:

• The most critical case must be taken in Eurocode 7. In this case maximum permissible load on the foundation is V = 494 kN/m (in Design Combination 2)

• This value is significantly higher than that allowed by the conventional design approach (V = 207 kN/m)

• Why?

Eurocode 7 [2]

• Eurocode 7 presents a significant change in philosophy compared to most previous codes.

• The preceding calculations are reliability based and ensure that the probability of collapse at the ULS is very low.

• The calculation says nothing about the SLS which must be addressed by a separate calculation. The F.O.S = 3 method implicitly addressed both SLS and ULS.

One reason can be related to the non-linearity of some limit

analysis calculations when checking geotechnical stability.

Consider the bearing capacity of a footing sitting on a cohesionless soil:

The exact solution is given by: V/B = 0.5BNWhere N must be computed numerically.

Why two calculations?

V

Why two calculations? [2]

N

25º 6.49

30º 14.8

35º 34.5

40º 85.6

45º 234

50º 743

Values of N are highly sensitive to the value of :


Example results

(unfactored) for B = 2m,

= 16 kN/m3:

V (kN/m) V/2750

40º 2750 1.0

38.4º 2030 1.35

34º 930 3.0

V


Active soil thrust for a

smooth wall P is given by:

P,

2

1 2HKPa

H

sin1

sin1a

K

Ka (DA1/1) Ka (DA1/2) ratio

30º 0.33 0.41 1.23

35º 0.27 0.34 1.27

40º 0.22 0.28 1.31

45º 0.17 0.23 1.35

50º 0.13 0.18 1.38

DA1/2 does not become critical

until '>45. However if soil

strength is relevant in other parts

of the problem (e.g. base sliding)

then DA1/2 typically dominates.


Factors of Safety and

overdesign factors

Factors of Safety

Many different definitions of factors of safety are used in geotechnical engineering. Three in common usage are listed below:

1. Factor on load.

2. Factor on material strength.

3. Factor defined as ratio of resisting forces (or moments) to disturbing forces (or moments).

The calculation process used to determine each of these factors for any given problem will in general result in a different failure mechanism, and a different numerical factor. Each FoS must therefore be interpreted according to its definition.

The Ultimate Limit State

• In general any given design is inherently stable and is nowhere near to its ultimate limit state.

• In order to undertake a ULS analysis it is necessary to drive the system to collapse by some means.

• This can be done implicitly or explicitly. In many conventional analyses the process is typically implicit. In a general numerical analysis it must be done explicitly.

Driving the system to ULS

There are three general ways to drive a system to ULS corresponding to the three FoS definitions previously mentioned:

1. Increasing an existing load in the system.

2. Reducing the soil strength

3. Imposing an additional load in the system

Computational Limit Analysis (including LimitState:GEO) solves problems using Method 1. However it can be straightforwardly programmed to find any of the other two types of Factors of Safety.

Method 1

Question: how much bigger does the load need to be to cause collapse, or, by what factor a does the load need to be increased to cause collapse:

This factor a is the „adequacy‟ factor.

Load * a

Method 2

Question: how much weaker does the soil need to be under the design load to cause collapse, or, by what factor F does the soil strength need to be reduced to cause collapse:

This factor F is the factor of safety on strength.

c/F, tan‟/F

Question: If the soil is failing around the structure, what is the ratio R of resisting forces to disturbing forces

This factor R = A / (P + S) is a factor of safety.

Method 3

Active earth

pressure

resultant A

Base friction S

Passive earth

pressure

resultant P

SLIDING

Method 3: analysis

Active

earth

pressure

resultant

ABase friction

S

Passive

earth

pressure

resultant P

Note that if R > 1:

• the passive earth pressure and base friction significantly exceed the active

earth pressure

• the system is therefore completely out of equilibrium.

• These assumed earth pressures are not possible without some external

disturbing agent.

Method 3: numerical analysis

• In a numerical analysis, equilibrium is required at all times.

• Therefore apply a „hypothetical‟ external force H in the direction of assumed

failure. Increase this force until failure occurs.

• Then determine ratio of other resisting to disturbing forces as before, but

ignore H itself.

• Mode of failure must be pre-determined in this method.

H

Equivalence of Methods

• At failure, Method 1( a = 1) and Method 2 (F = 1)

are identical

• Method 3 (R = 1) can be identical but only for the

given failure mechanism

EC7 DA1/1 EC7 DA1/2 EC7 DA2 EC7 DA3

Permanent

unfavourable load 1.35 1.0 1.351.0/1.35

Variable

unfavourable load 1.5 1.3 1.5 1.0/1.5

Permanent

favourable load 1.0 1.0 1.0 1.0

c' 1.0 1.25 1.0 1.25

tan' 1.0 1.25 1.0 1.25

cu 1.0 1.4 1.0 1.4

Resistance 1.0 1.0 1.1/1.4 N/A

Eurocode 7: DA1, DA2 and DA3

Eurocode Methods

• DA1/1 and DA2 are Method 3 approaches (also

Method 1 in simpler cases)

• Advantages:

• Familiar approach to geotechnical engineers for

straightforward problems

• Disadvantages:

• Much more challenging to conceptualise for complex

problems

• More difficult to analyse numerically

Action/Resistance approaches

• In Eurocode 7, retaining walls illustrate the

concept of an Action Effect.

• The action on the wall is a function not only of the applied

surface pressure, but the soil self weight and its strength.

Action/Resistance approaches [2]

EC7 DA1/1 EC7 DA2

Permanent

unfavourable

action

1.35 1.35

Variable

unfavourable

action

1.5 1.5

Permanent

favourable

action

1.0 1.0

Resistance 1.0 1.1/1.4

Unfavourable

Action (effect)

Resistance

Resistance

Actions and Resistances

Common source of confusion is to determine at what stage these factors

should be applied

2.4.2 Actions

NOTE Unfavourable (or destabilising) and favourable (or stabilising)

permanent actions may in some situations be considered as coming from

a single source. If they are considered so, a single partial factor may be

applied to the sum of these actions or to the sum of their effects.

Favourable/Unfavourable

1.35


Retaining Wall Design

Example

Problem specification

• An existing embankment is to be widened

• A stem wall is to be constructed and backfilled with granular material

• The widened embankment is to take additional loading.

• How wide (B) should the stem wall be?

5m

B

Φ = 27º, γ = 18kN/m3

Φ = 34º, γ = 18kN/m3

10kN/m2 10kN/m245kN/m2

Analysis

Rankine analysis using a virtual back is difficult due to

the varied surface loads and the inclined soil interface

Analysis [2]

Coulomb analysis more appropriate using a virtual back

and a range of wedge angles

LimitState:GEO Analysis

• Start with a DXF import of initial design


The future

Integrated structural – geotechnical design

Two storey frame on

slope

Integrated structural – geotechnical design

Serviceability Limit State

Eurocode 7: “2.4.8 (4) It may be verified that a sufficiently low

fraction of the ground strength is mobilised to keep

deformations within the required serviceability limits, provided

this simplified approach is restricted to design situations where:

• a value of the deformation is not required to check the

serviceability limit state;

• established comparable experience exists with similar

ground, structures and application method.”

This provides scope for the application of the „Mobilized

Strength Design‟ approach.

Serviceability Limit State [2]

• E.g. BS8002 utilizes a „mobilized‟ soil strength selected

such that ULS and SLS are simultaneously satisfied.

• The recommended factors of 1.2 on tan ’ and 1.5 on cu

are only applicable to soils that are medium dense or

firm (or stronger) for SLS to be satisfied.

• While the partial factors on soil strength seem similar to

those in Eurocode 7 DC2, the concept behind them is

completely different.

• The Eurocode 7 factors addresses ULS only, but could

potentially deal with SLS by „borrowing‟ the „mobilized

strength design‟ (MSD) approach which is the

conceptual basis of BS8002.


Conclusions

Conclusions (Limit State Design – EC7)

• Limit State Design aims to render the probability of a Limit state remote

• It may adopt a Material Factor approach (DA1/2) – this is easy to apply conceptually and to implement numerically

• It may adopt an Action or Action/Resistance Factor approach requiring care in application for more complex problems

• Adopting both approaches can protect against different forms of uncertainty

• EC7 requires distinct separate checks for ULS and SLS. This contrasts with some existing design codes which simultaneously deal with both ULS and SLS


Thank you for listeningAll analyses shown were run using LimitState:GEO

limitstate.com/geo

http://www.limitstate.com/geo

practical application of geotechnical limit analysis in limit state design

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